Given two distinct cuspidal automorphic $L$-functions (of $\mathrm{GL}_n$ and $\mathrm{GL}_m$ over $\mathbb{Q}$) one expects that their quotients will have infinitely poles, but this is surprisingly hard to prove. In this talk, I will discuss my recent work on the case $m=n-2$ and the primitivity of the $L$-functions of cuspidal automorphic $L$-functions of $\mathrm{GL}_3$. These methods also work for Artin $L$-functions and, more generally, for the $L$-functions of Galois representations under further hypotheses.